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Computational Social Choice - Course Overview and Key Concepts

Explore topics in computational social choice, Nash equilibria in games, and more in this course taught by Yair Zick in 2017 at NUS. Understand game theory basics, mixed Nash equilibria, and Von Neumann's Minimax Theorem.

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Computational Social Choice - Course Overview and Key Concepts

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  1. CS6283 Topics in Computer Science IV: Computational Social Choice Instructor: YairZick 2017

  2. Example: Allocating Goods Find an allocation that is: socially optimal? Envy free?

  3. Example: Facility Location • Need to place a shared facility that serves a bunch of people; placement rule (mechanism) needs to satisfy • High social welfare • Agents report their locations truthfully

  4. Example: Cost Sharing

  5. Example: Cost Sharing $25 $12

  6. This Course

  7. This Course • One lecture (2 hours SR@LT19) • Grade Breakdown: • 10% class participation (scribing, presentation) • 20% written assignments • 30% midterm (written test) • 40% final project

  8. Final Project

  9. Final Project

  10. Textbooks

  11. How to get an A in this class? Class participation – attend, ask questions! Prepare for class – review, readings, etc.… Start thinking about your project early!

  12. Other administrative Matters • Office hours – anytime, just email me! zick@comp.nus.edu.sg • Wednesdays 14:00-16:00 SR@LT19 • I will not be in on week 5 (Feb 8th) • We will schedule a makeup lesson on week 6, same time – need to agree on a day!

  13. Assignments and Exams You may work on assignments alone or in pairs, write who your partner was on the sheet. Submit assignments by hardcopy AND email Plagiarism on any task will be strictly dealt with…

  14. How about yourself? • Research interests? • What do you expect to achieve through this course?

  15. Non-Cooperative Games Nash equilibria and Game Theory Basics

  16. What is a Game? Players Actions (players can do something to affect the world) Preferences over outcomes

  17. Prisoner’s Dilemma – a classic puzzle • Two criminals are arrested. Interrogators not have enough evidence to convict them, but can convict them for a minor offense. • Each suspect is offered the same deal: • Implicate your friend, and we’ll let you go! • Both confess: get a sentence of 8 years • If one confesses and the other does not, the confessing party goes free while the other party serves 10 years. • Both stay quiet: both go to prison for one year(for the minor offense)

  18. Prisoner’s Dilemma – a classic puzzle What would you do?

  19. Nash Equilibria

  20. Normal Form Games A set of players Each player has a set of possible actions An action profile: a vector Player has utility from given by the value .

  21. Normal Form Games – Pure Nash Equilibria Given everyone else’s actions , the best response set of is An action profile is a (pure) Nash equilibrium if: “I’m doing the best I can, given everyone else’s actions!”

  22. Going to Dinner

  23. A Pure Nash Equilibrium… Always exists? Is poly-time computable? ? ? ? ? ? ? ? ? ? ?

  24. Example: ultimatum game Proposer offers an (integer) split of $10 Second player chooses to accept or reject What are the strategies? Nash equilibria? What do you think happens in practice? Which player has the advantage?

  25. Mixed Nash Equilibrium Playing a single strategy may be foolish – an opponent who knows you well can always beat you with a best response! It is often better to be unpredictable

  26. Mixed Nash Equilibrium Players are risk neutral! • Instead of choosing a single action, one can play a random mix of them • Player Utility: • Mixed NE: for all , • Unlike pure NE, a mixed NE always exists (by Brouwer’s Fixed Point Theorem)

  27. Example: zero-sum games Two player games For every pair of actions : For simplicity, we write

  28. Rock-Paper-Scissors

  29. In zero sum games: a Mixed Nash Equilibrium is poly-time computable? ? ? ? ? ? ? ? ? ? ?

  30. Maximin Strategy Assume w.l.o.g. that payoffs are non-negative Let is the maximin value; is the minimax value.

  31. The Minimax Theorem Von Neumann’s Minimax Theorem: maximin equals minimax.

  32. Reducing to rows/columns Proposition: Similarly:

  33. Reducing to rows/columns But since the other direction holds as well.

  34. LP Formulation

  35. Discussion Original proof by Von-Neumann was much more complicated (and also applied to cases where there are infinitely many players); proven before the discovery of LP duality! Game theory is a metaphor for real behavior – makes many implicit assumptions, tread lightly when applying it!

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